A158035 a(n) = 2*A158034(n) + 1, prime numbers p for which f = (2^p - 2^((p - 1) / 2 + 1) + 4p^2 - 8p) / (2p^2 - 2p) is an integer.

7, 23, 47, 167, 263, 359, 383, 479, 487, 503, 719, 839, 863, 887, 983, 1319, 1367, 1439, 1487, 1783, 1823, 2039, 2063, 2207, 2447, 2879, 2903, 2999, 3023, 3079, 3119, 3167, 3623, 3863, 4007, 4079, 4127, 4423, 4679, 4703, 4799, 4919, 5023, 5087, 5399, 5639

Offset: 1

Author

Reikku Kulon, Mar 11 2009

Keywords

• easy
• nonn

Comments

(p - 1) / 2 is often prime.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 112.

Crossrefs

Cf. A158034.

Cf. A002515 (Lucasian primes).

Cf. A145918 (exponential Sophie Germain primes).

Cf. A046318, A139876 (related to composite members of A158034: 243, 891, 1539, and 2511).

Programs

Maple

A158035 := proc(n) local i,am,p,tren;
am := [ ]:
for i from 2 to n do
  p := ithprime(i):
  tren := (2^(p) - 2^((p - 1) / 2 + 1) + 4*p^(2) - 8*p) / (2*p^(2) - 2*p):
  if (type( tren, 'integer') = 'true') then
    am := [op(am),p]:
  fi
od; RETURN(am) end:
A158035(740); # Jani Melik, May 06 2013

Mathematica

Select[Prime[Range[800]],IntegerQ[(2^#-2^((#-1)/2+1)+4#^2-8#)/(2#^2-2#)]&] (* Harvey P. Dale, Nov 08 2017 *)